Fast numerical schemes for Fredholm integral equations of the second kind have been developed. The integral equations are firstly discretized by the open Clenshaw-Curtis quadrature rule on a nodal point set $\Xi\sb{n}$. Generally n has to be chosen fairly large in order to obtain a certain accuracy. When the kernels are sufficiently smooth, we have shown that the linear systems of equations can be approximated well by their low rank approximations on $\Xi\sb{m}$ with $m \ll n$ by the eigenvalue expansions or the singular value decompositions of the integral operators. Most computations are now accomplished on $\Xi\sb{m}$. The Chebyshev expansions are used to define the interpolation formulas. We have shown that, if the kernel $\kappa\ \in\ C\sp{p}$ and the right hand side function $y\ \in\ C\sp{q}$ for some integers $p,q > 0$, the schemes converge at the rate of $o(1/m\sp{p-l}) + o (1/n\sp{\nu -1}$), where the integer $\nu \ge$ min(p,q).
When the kernels $\kappa(s,t$) are non-smooth along the line $s=t$, we have firstly described a high order quadrature rule. Then, we proposed two iteration methods to efficiently solve the corresponding equations.
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